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      <span class="sider-title">章节索引</span>
    </div>

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  <nz-content class="content-main" #contentMain>
    <div class="content-title">
      <h1>第五章 特征值与特征向量</h1>
      <p>总体介绍此项目的作用</p>
    </div>

    <!-- 各章节内容（省略，与之前相同） -->
    <section id="section1" class="content-section">
      <div class="text-content">
        <p class="text-subtitle">特征值与特征向量</p>
        <p class="text-theorem">
          <span class="text-theorem-title">特征值与特征向量的定义</span><br>
          &nbsp;&nbsp;&nbsp;&nbsp;设A是n阶矩阵，λ是一个数，若存在n 维非零列向量ξ,使得
          <app-formula formulaString="\overset{①}A\zeta = \lambda \zeta" [displayMode]="true"></app-formula>
          则称λ是A的特征值，ξ是A的对应于特征值λ的特征向量.
        </p>
        <p class="text-theorem">
          <span class="text-theorem-title">矩阵的特征值与特征向量的求法</span><br>
          &nbsp;&nbsp;&nbsp;&nbsp;由①式，得(λE-A)ξ=0, 因ζ≠0,故齐次方程组(λE-A)x=0零解，于是
          <app-formula formulaId="characteristic-polynomial" [displayMode]="true"></app-formula>
          &nbsp;&nbsp;&nbsp;&nbsp;②式称为A的特征方程，是未知量λ的n次方程 ，有n个根(重根按照重数计),λE-A称为特征矩阵，|λE-A|称为特征多项式.
          于是，求解具体型矩阵的特征值与特征向量，一般先用特征方程|λE-A|=0求出λ,再解齐次线性方程组(AE-A)x=0,求出特征向量.
        </p>
        <p class="text-theorem">
          <span class="text-theorem-title">特征值性质的重要结论</span>
          <app-formula formulaId="eigenvalue-properties" [displayMode]="true"></app-formula>
          <span class="text-theorem-title">特征向量性质的重要结论</span>
          <app-formula formulaId="eigen-vector-properties-full" [displayMode]="true"></app-formula>
          因此，可以得出一些常见的结论:
          <app-formula formulaId="eigen-table" [displayMode]="true"></app-formula>
        </p>
      </div>
    </section>
    <section id="section2" class="content-section">
      <div>
        <p class="text-subtitle">矩阵的相似</p>
        <p class="text-theorem">
          <span class="text-theorem-title">定义</span>
          &nbsp;&nbsp;&nbsp;&nbsp;设A, B是两个n阶方阵，若存在n阶可逆矩阵P,使得<app-formula formulaString="P^{-1}AP=B"></app-formula> ,则称A相似于B, 记成A～B.A、B相似.
        </p>
        <p class="text-theorem">
          <span class="text-theorem-title">相似矩阵的性质</span>
          &nbsp;&nbsp;&nbsp;&nbsp;<app-formula formulaString="(1)|A|=|B|"></app-formula> <br>
          &nbsp;&nbsp;&nbsp;&nbsp;<app-formula formulaString="(2)r(A)=r(B)"></app-formula> <br>
          &nbsp;&nbsp;&nbsp;&nbsp;<app-formula formulaString="(3)tr(A)=tr(B)"></app-formula> <br>
          &nbsp;&nbsp;&nbsp;&nbsp;<app-formula formulaString="(4)\lambda _{A} = \lambda _{B}"></app-formula> <br>
          &nbsp;&nbsp;&nbsp;&nbsp;<app-formula formulaString="(5)r(\lambda E-A) = r(\lambda E-B)"></app-formula> <br>
          &nbsp;&nbsp;&nbsp;&nbsp;<app-formula formulaString="(6)A, B的各阶主子式之和分别相等"></app-formula> <br>
        </p>
        <p class="text-theorem">
          <span class="text-theorem-title">相似矩阵的重要结论</span>
          &nbsp;&nbsp;&nbsp;&nbsp;<app-formula formulaString="(1)若A～B, 则A^{k}～B^{k}, f(A)～f(B)(其中f(x)是多项式)"></app-formula> <br>
          &nbsp;&nbsp;&nbsp;&nbsp;<app-formula formulaString="(2)若A～B, 且A可逆, 则A^{-1}～B^{-1}, f(A^{-1})～f(B^{-1})(其中f(x)是多项式)"></app-formula> <br>
          &nbsp;&nbsp;&nbsp;&nbsp;<app-formula formulaString="(3)若A～B, 则A^{*}～B^{*}"></app-formula> <br>
          &nbsp;&nbsp;&nbsp;&nbsp;<app-formula formulaString="(4)若A～B, 则A^{T}～B^{T}"></app-formula> <br>
          &nbsp;&nbsp;&nbsp;&nbsp;<app-formula formulaString="(5)若A～C, B～D, 则"></app-formula><app-formula formulaId="block-diagonal-similarity"></app-formula> <br>
          &nbsp;&nbsp;&nbsp;&nbsp;<app-formula formulaString="(6)A, B的各阶主子式之和分别相等"></app-formula> <br>
        </p>
      </div>
    </section>
    <section id="section3">
      <div>
        <p class="text-subtitle">矩阵的相似对角化</p>
        <p class="text-theorem">
          <span class="text-theorem-title">定义</span>
          &nbsp;&nbsp;&nbsp;&nbsp;设A, B是两个n阶方阵，若存在n阶可逆矩阵P,使得<app-formula formulaString="P^{-1}AP=B"></app-formula> ,其中A是对角矩阵, 则称A可相似对角化，记A～<app-formula formulaString="\mathcal{A}"></app-formula> , 称<app-formula formulaString="\mathcal{A}"></app-formula>是A的相似标准形.
        </p>
        <p class="text-theorem">
          <span class="text-theorem-title">矩阵可相似对角化的条件</span>
          &nbsp;&nbsp;&nbsp;&nbsp;由定义可知，若A 可相似对角化，即<app-formula formulaString="P^{-1}AP=\mathcal{A}"></app-formula> ,其中P可逆，等式两边同时左边乘P, 有 AP=P<app-formula formulaString="\mathcal{A}"></app-formula> ,记
          <app-formula formulaId="diagonalization" [displayMode]="true"></app-formula>
          则
          <app-formula formulaId="matrix-diagonalization-form" [displayMode]="true"></app-formula>
          即
          <app-formula formulaId="eigen-vector-transform" [displayMode]="true"></app-formula>
          也即
          <app-formula formulaId="eigen-equation-single" [displayMode]="true"></app-formula>
        </p>
        <p class="text-theorem">
          <span class="text-theorem-title">求可逆矩阵P，使得<app-formula formulaString="P^{-1}AP=B"></app-formula></span>&nbsp;&nbsp;&nbsp;&nbsp;在已知n阶矩阵A可相似对角化的条件下，其基本步骤如下.<br>
          &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<app-formula formulaString="(1)求A的特征值\lambda _{1}, \lambda _{2}, …, \lambda _{n}"></app-formula> <br>
          &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<app-formula formulaString="(2)求A的对应于特征值\lambda _{1}, \lambda _{2}, …, \lambda _{n}的线性无关的特征向量\zeta _{1}, \zeta _{2}, …, \zeta _{n}"></app-formula> <br>
          &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<app-formula formulaString="(3)令P=[\zeta _{1}, \zeta _{2}, …, \zeta _{n}], 则"></app-formula><app-formula formulaId="similarity-diagonalization"></app-formula> <br>
          需要注意的是P中特征向量ζ,与A中特征值λ,对应, 且P没有唯一性.
        </p>
        <p class="text-theorem">
          <span class="text-theorem-title">由特征值、特征向量反求A</span>
          &nbsp;&nbsp;&nbsp;&nbsp;若有可逆矩阵P,使得<app-formula formulaString="P^{-1}AP=\mathcal{A}"></app-formula> ,则<app-formula formulaString="A=P\mathcal{A}P^{-1}"></app-formula> , 这是反求A的一个基本思路.
          <app-formula formulaId="diagonalization-2d"></app-formula>
        </p>
        <p class="text-theorem">
          <span class="text-theorem-title"><app-formula formulaString="A^{k}及f(A)"></app-formula> </span>
          <app-formula formulaId="diagonalization-power-function"></app-formula>
        </p>
      </div>
    </section>
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